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Efficient Dense Linear Algebra Solvers in Parallel Computing

Learn efficient methods like Gaussian elimination with pivoting for solving dense linear equations in parallel computing. Understand GE algorithms, LU factorization, matrix operations, and dealing with numerical stability issues.

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Efficient Dense Linear Algebra Solvers in Parallel Computing

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  1. CS 240A: Solving Ax = b in parallel • Dense A: Gaussian elimination with partial pivoting • Same flavor as matrix * matrix, but more complicated • Sparse A: Iterative methods – Conjugate gradient etc. • Sparse matrix times dense vector • Sparse A: Gaussian elimination – Cholesky, LU, etc. • Graph algorithms • Sparse A: Preconditioned iterative methods and multigrid • Mixture of lots of things

  2. Dense Linear Algebra (Excerpts) James Demmel http://www.cs.berkeley.edu/~demmel/cs267_221001.ppt

  3. Motivation • 3 Basic Linear Algebra Problems • Linear Equations: Solve Ax=b for x • Least Squares: Find x that minimizes S ri2where r=Ax-b • Eigenvalues: Findland x where Ax = l x • Lots of variations depending on structure of A (eg symmetry) • Why dense A, as opposed to sparse A? • Aren’t “most” large matrices sparse? • Dense algorithms easier to understand • Some applications yields large dense matrices • Ax=b: Computational Electromagnetics • Ax = lx: Quantum Chemistry • Benchmarking • “How fast is your computer?” = “How fast can you solve dense Ax=b?” • Large sparse matrix algorithms often yield smaller (but still large) dense problems

  4. Review of Gaussian Elimination (GE) for solving Ax=b • Add multiples of each row to later rows to make A upper triangular • Solve resulting triangular system Ux = c by substitution … for each column i … zero it out below the diagonal by adding multiples of row i to later rows for i = 1 to n-1 … for each row j below row i for j = i+1 to n … add a multiple of row i to row j for k = i to n A(j,k) = A(j,k) - (A(j,i)/A(i,i)) * A(i,k)

  5. Refine GE Algorithm (1) • Initial Version • Remove computation of constant A(j,i)/A(i,i) from inner loop … for each column i … zero it out below the diagonal by adding multiples of row i to later rows for i = 1 to n-1 … for each row j below row i for j = i+1 to n … add a multiple of row i to row j for k = i to n A(j,k) = A(j,k) - (A(j,i)/A(i,i)) * A(i,k) for i = 1 to n-1 for j = i+1 to n m = A(j,i)/A(i,i) for k = i to n A(j,k) = A(j,k) - m * A(i,k)

  6. Refine GE Algorithm (2) • Last version • Don’t compute what we already know: zeros below diagonal in column i for i = 1 to n-1 for j = i+1 to n m = A(j,i)/A(i,i) for k = i to n A(j,k) = A(j,k) - m * A(i,k) for i = 1 to n-1 for j = i+1 to n m = A(j,i)/A(i,i) for k = i+1 to n A(j,k) = A(j,k) - m * A(i,k)

  7. Refine GE Algorithm (3) • Last version • Store multipliers m below diagonal in zeroed entries for later use for i = 1 to n-1 for j = i+1 to n m = A(j,i)/A(i,i) for k = i+1 to n A(j,k) = A(j,k) - m * A(i,k) for i = 1 to n-1 for j = i+1 to n A(j,i) = A(j,i)/A(i,i) for k = i+1 to n A(j,k) = A(j,k) - A(j,i) * A(i,k)

  8. Refine GE Algorithm (4) • Last version for i = 1 to n-1 for j = i+1 to n A(j,i) = A(j,i)/A(i,i) for k = i+1 to n A(j,k) = A(j,k) - A(j,i) * A(i,k) • Split Loop for i = 1 to n-1 for j = i+1 to n A(j,i) = A(j,i)/A(i,i) for j = i+1 to n for k = i+1 to n A(j,k) = A(j,k) - A(j,i) * A(i,k)

  9. Refine GE Algorithm (5) for i = 1 to n-1 for j = i+1 to n A(j,i) = A(j,i)/A(i,i) for j = i+1 to n for k = i+1 to n A(j,k) = A(j,k) - A(j,i) * A(i,k) • Last version • Express using matrix operations (BLAS) for i = 1 to n-1 A(i+1:n,i) = A(i+1:n,i) * ( 1 / A(i,i) ) A(i+1:n,i+1:n) = A(i+1:n , i+1:n ) - A(i+1:n , i) * A(i , i+1:n)

  10. What GE really computes • Call the strictly lower triangular matrix of multipliers M, and let L = I+M • Call the upper triangle of the final matrix U • Lemma (LU Factorization): If the above algorithm terminates (does not divide by zero) then A = L*U • Solving A*x=b using GE • Factorize A = L*U using GE (cost = 2/3 n3 flops) • Solve L*y = b for y, using substitution (cost = n2 flops) • Solve U*x = y for x, using substitution (cost = n2 flops) • Thus A*x = (L*U)*x = L*(U*x) = L*y = b as desired for i = 1 to n-1 A(i+1:n,i) = A(i+1:n,i) / A(i,i) A(i+1:n,i+1:n) = A(i+1:n , i+1:n ) - A(i+1:n , i) * A(i , i+1:n)

  11. Problems with basic GE algorithm • What if some A(i,i) is zero? Or very small? • Result may not exist, or be “unstable”, so need to pivot • Current computation all BLAS 1 or BLAS 2, but we know that BLAS 3 (matrix multiply) is fastest (earlier lectures…) for i = 1 to n-1 A(i+1:n,i) = A(i+1:n,i) / A(i,i) … BLAS 1 (scale a vector) A(i+1:n,i+1:n) = A(i+1:n , i+1:n ) … BLAS 2 (rank-1 update) - A(i+1:n , i) * A(i , i+1:n) Peak BLAS 3 BLAS 2 BLAS 1

  12. Pivoting in Gaussian Elimination • A = [ 0 1 ] fails completely, even though A is “easy” • [ 1 0 ] • Illustrate problems in 3-decimal digit arithmetic: • A = [ 1e-4 1 ] and b = [ 1 ], correct answer to 3 places is x = [ 1 ] • [ 1 1 ] [ 2 ] [ 1 ] • Result of LU decomposition is • L = [ 1 0 ] = [ 1 0 ] … No roundoff error yet • [ fl(1/1e-4) 1 ] [ 1e4 1 ] • U = [ 1e-4 1 ] = [ 1e-4 1 ] … Error in 4th decimal place • [ 0 fl(1-1e4*1) ] [ 0 -1e4 ] • Check if A = L*U = [ 1e-4 1 ] … (2,2) entry entirely wrong • [ 1 0 ] • Algorithm “forgets” (2,2) entry, gets same L and U for all |A(2,2)|<5 • Numerical instability • Computed solution x totally inaccurate • Cure: Pivot (swap rows of A) so entries of L and U bounded

  13. Gaussian Elimination with Partial Pivoting (GEPP) • Partial Pivoting: swap rows so that each multiplier • |L(i,j)| = |A(j,i)/A(i,i)| <= 1 for i = 1 to n-1 find and record k where |A(k,i)| = max{i <= j <= n} |A(j,i)| … i.e. largest entry in rest of column i if |A(k,i)| = 0 exit with a warning that A is singular, or nearly so elseif k != i swap rows i and k of A end if A(i+1:n,i) = A(i+1:n,i) / A(i,i) … each quotient lies in [-1,1] A(i+1:n,i+1:n) = A(i+1:n , i+1:n ) - A(i+1:n , i) * A(i , i+1:n) • Lemma: This algorithm computes A = P*L*U, where P is a • permutation matrix • Since each entry of |L(i,j)| <= 1, this algorithm is considered • numerically stable • For details see LAPACK code at www.netlib.org/lapack/single/sgetf2.f

  14. Converting BLAS2 to BLAS3 in GEPP • Blocking • Used to optimize matrix-multiplication • Harder here because of data dependencies in GEPP • Delayed Updates • Save updates to “trailing matrix” from several consecutive BLAS2 updates • Apply many saved updates simultaneously in one BLAS3 operation • Same idea works for much of dense linear algebra • Open questions remain • Need to choose a block size b • Algorithm will save and apply b updates • b must be small enough so that active submatrix consisting of b columns of A fits in cache • b must be large enough to make BLAS3 fast

  15. Blocked GEPP (www.netlib.org/lapack/single/sgetrf.f) for ib = 1 to n-1 step b … Process matrix b columns at a time end = ib + b-1 … Point to end of block of b columns apply BLAS2 version of GEPP to get A(ib:n , ib:end) = P’ * L’ * U’ … let LL denote the strict lower triangular part of A(ib:end , ib:end) + I A(ib:end , end+1:n) = LL-1 * A(ib:end , end+1:n)… update next b rows of U A(end+1:n , end+1:n ) = A(end+1:n , end+1:n ) - A(end+1:n , ib:end) * A(ib:end , end+1:n) … apply delayed updates with single matrix-multiply … with inner dimension b

  16. Overview of LAPACK • Standard library for dense/banded linear algebra • Linear systems: A*x=b • Least squares problems: minx || A*x-b ||2 • Eigenvalue problems: Ax = lx, Ax = lBx • Singular value decomposition (SVD): A = USVT • Algorithms reorganized to use BLAS3 as much as possible • Basis of math libraries on many computers, Matlab 6 • Many algorithmic innovations remain • Automatic optimization • Quadtree matrix data structures for locality • New eigenvalue algorithms

  17. Parallelizing Gaussian Elimination • Recall parallelization steps from earlier lecture • Decomposition: identify enough parallel work, but not too much • Assignment: load balance work among threads • Orchestrate: communication and synchronization • Mapping: which processors execute which threads • Decomposition • In BLAS 2 algorithm nearly each flop in inner loop can be done in parallel, so with n2 processors, need 3n parallel steps • This is too fine-grained, prefer calls to local matmuls instead • Need to discuss parallel matrix multiplication • Assignment • Which processors are responsible for which submatrices? for i = 1 to n-1 A(i+1:n,i) = A(i+1:n,i) / A(i,i) … BLAS 1 (scale a vector) A(i+1:n,i+1:n) = A(i+1:n , i+1:n ) … BLAS 2 (rank-1 update) - A(i+1:n , i) * A(i , i+1:n)

  18. Different Data Layouts for Parallel GE (on 4 procs) Load balanced, but can’t easily use BLAS2 or BLAS3 Bad load balance: P0 idle after first n/4 steps Can trade load balance and BLAS2/3 performance by choosing b, but factorization of block column is a bottleneck The winner! Complicated addressing

  19. Review: BLAS 3 (Blocked) GEPP for ib = 1 to n-1 step b … Process matrix b columns at a time end = ib + b-1 … Point to end of block of b columns apply BLAS2 version of GEPP to get A(ib:n , ib:end) = P’ * L’ * U’ … let LL denote the strict lower triangular part of A(ib:end , ib:end) + I A(ib:end , end+1:n) = LL-1 * A(ib:end , end+1:n)… update next b rows of U A(end+1:n , end+1:n ) = A(end+1:n , end+1:n ) - A(end+1:n , ib:end) * A(ib:end , end+1:n) … apply delayed updates with single matrix-multiply … with inner dimension b BLAS 3

  20. Review: Row and Column Block Cyclic Layout processors and matrix blocks are distributed in a 2d array pcol-fold parallelism in any column, and calls to the BLAS2 and BLAS3 on matrices of size brow-by-bcol serial bottleneck is eased need not be symmetric in rows and columns

  21. Distributed GE with a 2D Block Cyclic Layout block size b in the algorithm and the block sizes brow and bcol in the layout satisfy b=brow=bcol. shaded regions indicate busy processors or communication performed. unnecessary to have a barrier between each step of the algorithm, e.g.. step 9, 10, and 11 can be pipelined

  22. Distributed GE with a 2D Block Cyclic Layout

  23. Matrix multiply of green = green - blue * pink

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